Download Estimating the tolerance of species to the effects

arXiv:1308.3584v1 [q-bio.PE] 16 Aug 2013
Estimating the tolerance of species to the effects of
global environmental change
Serguei Saavedra∗†, Rudolf P. Rohr†,
Vasilis Dakos and Jordi Bascompte
Integrative Ecology Group
Estación Biológica de Doñana, EBD-CSIC
Calle Américo Vespucio s/n, E-41092 Sevilla, Spain
Two-sentence summary: Global environmental change is affecting the strength
of interspecific interactions. The authors here estimate how much change species can
tolerate before becoming extinct, and find that species tolerance is very sensitive to the
net direction of change.
∗
†
To whom correspondence should be addressed. E-mail: [email protected]
These authors contributed equally to this work
1
Global environmental change is affecting species distribution and
their interactions with other species. In particular, the main drivers
of environmental change strongly affect the strength of interspecific
interactions with considerable consequences to biodiversity. However, extrapolating the effects observed on pair-wise interactions to
entire ecological networks is challenging. Here we propose a framework to estimate the tolerance to changes in the strength of mutualistic interaction that species in mutualistic networks can sustain
before becoming extinct. We identify the scenarios where generalist
species can be the least tolerant. We show that the least tolerant
species across different scenarios do not appear to have uniquely
common characteristics. Species tolerance is extremely sensitive to
the direction of change in the strength of mutualistic interaction, as
well as to the observed mutualistic trade-offs between the number
of partners and the strength of the interactions.
2
Introduction
Global environmental change is accelerating as anthropogenic effects are increasing over
short time scales.1, 2 The effects of such unprecedented change are modifying the abundance, physiology, and geographic range of individual species,3, 4, 5 and also affecting
species interactions with the potential to modify ecosystem services, such as biological
control and pollination.6, 7
Empirical studies of mutualistic systems6 suggest that the main drivers of global environmental change are nitrogen enrichment, increase in CO2 , and habitat fragmentation,
among others. Importantly, these drivers can alter the frequency of pollinator visits to
flowering plants due to climate-induced phenological shifts, nectar variability, or a decrease in flowers’ abundance.6 For example, it has been shown that solitary bees exhibit a
small foraging range so that pollinator visitation can decrease if foraging distances become
larger after habitat fragmentation.8 Similarly, it has been shown that land fragmentation
can lead to a significant increase of pollinator visitation in a community of flower-visiting
insects in red clover.9 In general, empirical evidence has shown that the effects of global
environmental change can either increase or decrease the strength of mutualistic interaction, with the majority of cases showing the latter.6
While few studies have looked at the association of the effects of global environmental
change with the loss of mutualistic interactions and community persistence,7 species’
tolerance to these effects have been restricted to pair-wise interactions.6 Thus, it is still
unknown the degree to which these effects will scale all the way up to entire networks of
interactions, and which species would face a higher extinction risk.
On the theoretical front, recent work on ecological networks has shown important architectural properties that can facilitate species coexistence.10, 11, 12 Focusing on individual
3
species, research has shown that species’ generalization level13 (i.e., its number of interactions, or degree) and their contribution to the nested architecture of the network14 play
a key role for their survival. This work, however, has assumed constant environmental
conditions.
Here, we introduce a theoretical framework to estimate species’ tolerance to the effects
of global environmental change. We focus on the association of species’ tolerance with
their level of generalization and their contribution to network architecture. To obtain a
mechanistic understanding of the range of species’ tolerance, we start by applying our
framework to a 3-species community. We then move to study species’ tolerance in large
communites. In general, we find that endangered species do not have unique characteristics. Importantly, our findings reveal that in order to estimate species’ tolerance, first,
one needs to identify the net effect of the observed environmental change.
Results
A small mutualistic community
We start by applying our framework (Methods) to a 3-species community (one plant
and two pollinators with different strengths of mutualistic interaction represented by the
width of links in Fig. 1a). Figures 1a and 1b show the community moving gradually
from a weak to a strong mutualism, and from a strong to a weak mutualism, respectively.
Note that in a weak mutualism regime, competition effects are stronger than mutualistic
effects, whereas the opposite occurs in a strong mutualism regime. This implies that a
species’ initial abundance and intrinsic growth rate can be different depending on whether
we start from a weak or a strong mutualism regime. We quantify a species’ tolerance as
the change in strength of mutualistic interaction that species can sustain before becoming
extinct.
4
Under initial conditions, all species co-exist, but as soon as we start changing the
strength of mutualistic interaction γo , species’ initial abundances also begin to change
(Fig. 1). While gradual changes in the strength of mutualistic interaction can increase
the abundances of the plant and one of the pollinators, the interspecific competition
between pollinators can drive one of the two into extinction. Interestingly, pollinators’
tolerance is not the same under both directions of change. One pollinator (red) goes
extinct when moving from weak to strong mutualism, but it survives when moving from
strong to weak mutualism. The other pollinator (blue) presents the opposite pattern.
This suggests that species’ tolerance can be extremely sensitive to both the direction of
change in the strength of mutualistic interaction and the variability of this strength across
species.
To scale up from small to large communities, we apply our framework to 59 pollination
and seed-dispersal networks (Methods). Figure 2 shows the species’ tolerance (node color)
of each of the 80 plants and 97 pollinators belonging to one network located in the high
temperate Andes of central Chile.15 The darker the color, the higher the change in the
strength of mutualistic interaction sustained by each species before becoming extinct and,
in turn, the stronger its tolerance. In this example, changes are introduced by moving
from a weak to a strong mutualism with a low mutualistic trade-off between a species’
number and strength of interactions (Methods).
Figure 2 also shows the degree (node size) and contribution to nestedness (node position) for each of the species. Degree is given by the initial number of mutualistic interactions, while contribution to nestedness is measured by the extent to which those
interactions contribute to the nested architecture of the network relative to an expected
contribution (Methods). These two measures are very weakly correlated, providing almost
independent information.14 Surprisingly, this figure reveals that some specialist species
5
can be more tolerant than generalist species (far right). In the next sections, we explore
how general this observation is and under what scenario is most likely to be found. We
first consider the case where the effects of global change increase the strength of mutualistic interaction, followed by the reverse case where global change weakens the strength
of mutualistic interaction, which seems to be the most likely case in nature.6
Increasing mutualism
It has been shown that, under constant environmental conditions, generalists and weak
contributors to nestedness can have the highest chances of survival.11, 14, 13 To test if this
observation holds under an increase in the strength of mutualistic interaction (Methods),
we use Spearman rank correlation to measure and properly compare across all networks
the association of a species’ tolerance with its degree and its contribution to nestedness.
We find positive and significant correlations in both cases (Figs. 3a-3c). Interestingly,
the positive association between species’ tolerance and degree does not hold when the
trade-off between the number of interactions and benefits received is large (Fig. 3c). In
general, these results reveal that generalists and strong contributors to nestedness are the
most tolerant of an increase in the strength of mutualistic interaction.
To further explore the magnitude of these associations, we calculate the ratio of the
correlation norms d of nestedness with species’ tolerance and degree with species’ tolerance across the 59 networks (Methods). The ratio measures how comparable the two
correlations are in the magnitudes. A value of d > 1 corresponds to the case where
contribution to nestedness has on average larger correlations with species’ tolerance than
degree, and vice versa for values of d < 1 (Fig. 3). We find a ratio of d = 1.16 when
the mutualistic trade-off is small (Fig. 3b), and ratios d < 1 in the other two scenarios.
This suggests that degree is not always the best estimator of species’ tolerance. This also
6
suggests that under a small level of mutualistic trade-off, the nested organization of the
network can have the highest influence on the persistence of the community.
Decreasing mutualism
More typically, however, global environmental change is expected to weaken the strength
of interactions in mutualistic networks6 (Methods). Surprisingly, we find the opposite
patterns of what we found when moving from a weak to a strong mutualism. Here,
generalists are highly tolerant with small and large mutualistic trade-offs (Fig. 3e-3f),
while strong contributors to nestedness are the least tolerant in the three scenarios (Fig.
3d-3f).
Importantly, we find that with none and small mutualistic trade-offs (Fig. 3d-3e),
species’ tolerance is more strongly associated with their contribution to nestedness than
with degree, i.e., d > 1. These results, together with the ones found when increasing
mutualism, show that the least tolerant species across different scenarios do not appear to
have uniquely common characteristics. Instead, our results reveal that species’ tolerance
depends on both the direction of change and the mutualistic trade-offs.
Discussion
In this paper, we have introduced a theoretical framework to estimate species’ tolerance
to change in the strength of mutualistic interaction as a potential effect of global environmental change. We have analyzed the extent to which both the direction of change
and the mutualistic trade-offs are associated with species’ tolerance. The former was
investigated by moving from weak to strong mutualism and vice versa. The latter was
investigated by modulating an observed mutualistic trade-off between a species’ number
and strength of interactions. We have found that specific combinations of direction of
7
change and mutualistic trade-offs can have a different impact on species’ tolerance.
Contrary to the scenario of constant environmental conditions, where degree is the
gold standard measure for estimating species’ tolerance, here we have demonstrated that
in a changing environment this is not always the case. In fact, consistent with empirical
observations,7, 16 generalist species can be the most vulnerable. We have found that generalists are the least tolerant under two scenarios: when the effect of global environmental
change strengthens mutualism with a large mutualistic trade-off; and when the effect of
global environmental change weakens mutualism with a very small mutualistic trade-off.
This suggests that the tolerance of generalist species needs to be estimated relative to
the direction of change, as well as to the mutualistic trade-off affecting specialist and
generalist species. This is important since generalist species can have significant implications for the long-term functioning of ecosystems.17, 18 Moreover, we have found that
under half of the observed scenarios, species’ tolerance can be more strongly associated
with their contribution to nestedness than with degree. In general, this reveals that the
least tolerant species across different scenarios do not appear to have uniquely common
characteristics.
Because many complex systems are facing systemic risk, the results presented here are
not only of relevance in ecology but beyond.19, 20, 21 For example, in socio-economic systems, strong contributors to nestedness can be the most vulnerable to fail.14 Within our
framework, this could be explained by a possible decoupling of cooperative interactions
when moving from strong to weak mutualism and a lack of adaptation to these new conditions. Similar results can be observed in the banking sector, where financial institutions
around the world are strengthening or weakening their cooperative interactions.22, 23 A
valuable lesson from our results is that a node in these cooperative networks is never too
big, too connected, or too peripherial to fail.
8
As new studies continue to confirm a dramatic loss of pollination systems around the
world,24, 25, 7, 16 it is becoming increasingly important to properly identify the characteristics of vulnerable species in these networks. Our findings reveal that in order to estimate
species’ tolerance to change, first, one needs to identify the net effect of environmental
change, which depends on both species-level and network-level properties. Specifically,
both the direction of change in the strength of mutualistic interaction and the mutualistic
trade-off highly modulate the ranking of species in terms of their tolerance to the effects
of global environmental change.
Materials and Methods
Data. We investigate species’ tolerance over 59 mutualistic networks that were located at
different abiotic conditions around the world. This dataset is published in Rezende et al.18
Framework. In the text below, we describe in detail our proposed framework to study
species’ tolerance to the effects of global environmental change.
Species’ tolerance. We quantify a species’ tolerance as the change in the strength
of mutualistic interaction that it can sustain before becoming extinct.
Model. We model the dynamics of mutualistic systems composed of a set of plants
and a set of animals (indicated by the upper indices (A) and (P)) using the same set of
differential equations as in Bastolla et al.:10



dSi
dt


dSi
dt
(P )
(A)
P
(P )
(P )
−
P
j βij Sj
(P )
(P )
+
(A)
(A)
−
P
(A)
(A)
+
= Si (αi
= Si (αi
j βij Sj
9
(P )
(A)
γij Sj
P (P ) (A) )
1+h j γij Sj
P (A) (P )
j γij Sj
P (A) (P ) )
1+h j γij Sj
j
(1)
The equations for pollinator populations can be written in a symmetric form interchanging the indices (P ) and (A). Since there is no data to fully parametrize our dynamical system with meaningful biological information, we use a mean field approximation10
for the competition term (i.e., βii = 1 and βij = 0.2 if i 6= j) and we set the handling time
h = 0.1. While these can be taken as arbitrary values, we find that our main conclusions
are robust to the choice of different parameters.
The variables Si denote the abundance of species i. The parameter αi represents the
intrinsic growth rate, and γij denotes the strength of the mutualistic interaction between
plants and animals. Simulations are performed by integrating the system of ordinary
differential equations using the Matlab routline ode45. Species are considered extinct
when their abundance density Si is lower than 100 times the machine precision.
Consistent with empirical observations,26, 27 initial abundances are set proportional to
the number of interactions. We initialize all growth rates such that the initial abundances
are at a feasible and stable equilibrium (see below). Finally, since we assume that species
have no time to adapt,3 all surviving species preserve their initial growth rates, interspecific competition and asymmetric benefits through the simulations.
Mutualistic trade-off. Consistent with field observations,28, 29 we generalize the soft
mean field approximation of Bastolla et al.10 by introducing explicitly a trade-off between mutualistic strength and species degree, which modulates the mutualistic trade-off
between generalists and specialists:
γij =
γo yij
,
kiδ
(2)
where γo represent the basal level of mutualistic strength, ki the degree of species i, and
10
yij = 1 if species i and j interact and zero otherwise. The parameter δ modulates the
trade-off. The higher the trade-off, the higher the strength (mutualistic benefit) perceived
by specialists. In our simulations, we consider δ = 0, δ = 0.5, and δ = 2 for none, small
(sub-linear), and large (super linear) mutualistic trade-offs, respectively. Note that the
case δ = 0 is equivalent to the soft mean field approximation used in Bastolla et al.10
Direction of change. Changes in the strength of mutualistic interaction are modeled by either increasing or decreasing the mutualistic strength γo in the dynamic model
described above. The initial and final γo for the increasing direction are set, respectively,
to γo = 0 and γo = 10τ . Similarly, the initial and final γo for the decreasing direction
are set, respectively, to γo = 3τ and γo = 0. Here, τ is the analytical threshold at which
each network changes from a weak to a strong regime (see below). Changes to γo are
introduced in small steps when reaching a new equilibrium of abundances. We find no
significant differences if changes in γo are introduced at any point of our simulations.
During any new simulation step, species’ initial abundances are the final abundances of
the previous step.
Weak and strong mutualism. By definition, a mutualistic system is in a weak regime,
if and only if the following 2 × 2 block matrix
β (P ) −γ (P )
M=
−γ (A) β (A)
(3)
is positive definite (i.e., all eigenvalues of M + M T are positive). If that condition is not
satisfied, then we say that the system is in a strong regime. This definition is a generalization to the non-symmetric case of the weak/strong concept introduced in Bastolla et al.10
11
Intuitively, being in a weak regime means that mutualistic interactions are “weaker” than
competitive interactions. In the case of a fully-connected network without interspecific
competition (βij = 0 for i 6= j) and the same strength of mutualistic interaction between
all pairs of species, this condition is equivalent to the inequality derived in Bascompte et
al.30 The condition of being in the weak regime is a stronger stability condition than the
usual one based on the eigenvalues of the Jacobian matrix evaluated at an equilibrium
point. Under the weak condition, any feasible equilibrium point (i.e., strictly positive
abundance values that vanish the right side of the model equation) is automatically globally stable since it is possible to construct a Lyapunov function.31 Then, when the system
enters the strong mutualism regime, with a handling time of h = 0, non-trivial fixed points
are not any more granted to be stable. This means that for a large enough γij the system
blows up. The only way to recover the stability of an equilibrium point in the strong
regime is, first to have a positive handling time h > 0, and second that the abundances
at the equilibrium point are large enough such that the system is locally stable (around
that equilibrium point). In our framework, the transition from weak to strong mutualism
is simply computed as a threshold of the basal level of mutualistic strength γo . Given a
network, competition parameter values, and a trade-off value δ, we can find a positive
threshold, called τ > 0, such that if γo < τ the system is in the weak regime, and if γo ≥ τ
the system is in the strong regime. Note that this threshold is network, competition parameters, and trade-off dependent. For the same value of γo , a given network may be in
the strong regime while another network can be in the weak regime.
Initialization of weak to strong mutualism. For our simulations from weak to
strong mutualism, we range the mutualistic strength γo from 0 to 10 times the threshold
τ , i.e., γo ∈ [0, . . . , 10τ ]. We initialize the system such that at γo = 0 it is at a globally
12
stable equilibrium where all species have a positive abundance. Consistent with field
observations,26, 27 we choose this equilibrium point such that the abundances are proportional to the species degree ki and the total abundances of each species guild is equal to
P
5, i.e., Si (γo = 0) = 5ki / i ki . Note that the global stability of this potential feasible
equilibrium is granted as we are in the weak regime. Finally, to ensure that this feasible
point is also an equilibrium point, we have to choose the intrinsic growth rate such that
αi =
X
βij Sj (γo = 0).
(4)
j
Initialization of strong to weak mutualism. For our simulations from strong to
weak mutualism, we range the mutualistic strength γo from 0 to 3 times the threshold
τ , i.e., γo ∈ [0, . . . , 3τ ]. We initialize the system such that at γo = 3τ it is at a stable
equilibrium where all species have positive abundances. As above, we also choose this
equilibrium point such that the abundances are proportional to the species degree ki , i.e.,
P
Si (γo = 3τ ) = So ki / i ki . However, since we are in the strong regime, this equilibrium
point may not be stable for any value of the total abundance So . In particular, for a
very low value of So , the equilibrium point may be unstable, and we only can recover its
stability when So has crossed a threshold. Then, the first task is to find this threshold
in total abundance. For that, we first have to linearize the right side of the equations
system. The linearized system around equilibrium abundances Ŝi is given by:



(A)
where α̃i
=
P
(A)
j
(P )
dSi
dt
(A)
dSi
dt
(A)
βij Ŝi
(P )
(P )
= Si (α̃i
−
P
(A)
−
P
(A)
= Si (α̃i
−
P
j
(P )
(A)
(A)
j βij Sj
(P )
j
(P )
βij Sj
γij Ŝj /(1 + h
13
P
j
+
P
+
P
j
(P )
(A)
(A)
(P )
γ̃ij Sj )
(5)
j γ̃ij Sj ),
γij Ŝj )2 and the linearized strength of
mutualistic interaction γ̃ij = γij /(1 + h
P
j
γij Ŝj )2 (similar expressions hold for plants).
Then, for the linearized system we can extract the linearized version of the M matrix:
β (P ) −γ̃ (P )
M̃ =
−γ̃ (A) β (A)
(6)
Note that the elements of M̃ are functions of the abundance values around which we
have linearized the dynamical system. Our stability condition is that M̃ , the linearized
version of M , is positive definite, i.e., the system is locally stable in the weak regime. In
our framework, we need to find which value of total abundance So , with Ŝi = Si (γo = 3τ ),
makes the matrix M̃ be positive definite. We choose this exact value as starting point
for the total abundance of the feasible point. Finally, to make that feasible point an
equilibrium point, we have to choose the intrinsic growth rate such that
αi =
X
P
βij Sj (γo = 3τ ) −
j
γij Sj (γo = 3τ )
P
1 + h j γij Sj (γo = 3τ )
j
(7)
Contribution to nestedness. Individual contribution to nestedness for each species
or node quantifies the degree to which nestedness compares with the same value when
randomizing just the interactions of that particular node.14 In calculating nestedness contributions, the interactions of a node are randomized according to the null model specified
in Bascompte et al.;32 we used 1000 random replicates. Here, nestedness is quantified using the measure proposed in Bastolla et al.,10 which is analytically linked to the dynamics
of the mutualistic model. Other measures of nestedness and null models yield the same
general results for the species-level analysis.14
Ratio of the norms. The ratio d of the correlation norms between contribution to nest-
14
edness and degree, x and y, is defined as d = |x|/|y|, where |x| =
pPm
i
x2i , |y| =
pPm
i
yi2 .
Here, xi and yi correspond to the Spearman rank correlations for each of the m = 59 observed networks. The ratio d provides a measure of the relative length of the correlations
between contribution to nestedness and degree in an m−dimensional space. Typically,
values within 0.9 < d < 1.1 are considered significantly similar. The same general results
are obtained if we use species’ sum of the strength of mutualistic interaction instead of
their number of interactions. Similarly, we find no significant association of the observed
Spearman rank correlations with network connectance or size, confirming the comparability of our results across networks.
ACKNOWLEDGMENTS We thank Alex Arenas and Jason Tylianakis for insightful discussions. Funding was provided by the European Research Council through an
Advanced Grant (JB), CONACYT (SS), FP7-REGPOT-2010-1 program under project
264125 EcoGenes (RPR), and Rubicon grant NWO and a Marie Curie IEF-EU fellowship
(VD).
Author contributions All authors contributed extensively to the work presented in
this paper.
Competing financial interests The authors declare no competing financial interests.
15
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18
equilibrium abundance
10
8
Plant
8
Pollinator
6
6
4
4
2
0
0
(weak)
threshold γ
b 10
threshold γ
a
2
extinction
extinction
2
4
6
mutualistic strength γ
8
10
(strong)
0
0
(weak)
2
4
6
mutualistic strength γ
8
Figure 1: Pollinators tolerance depends on the direction of change in the strength of mutualistic interaction. The figure plots the equilibrium abundances of a 3-species community (one plant and two
pollinators) in the face of A an increase and B a decrease in the strength of mutualistic interaction. We
set different strengths for each of the interspecific interactions (represented by the width of links). One
pollinator (red) goes extinct when increasing the mutualistic interaction strength, but it survives when
decreasing mutualism. The other pollinator (blue) presents the opposite pattern. The dashed line corresponds to the value of γo at which the community shows a transition between weak and strong mutualism
(Methods). In all our simulations we start at a feasible and stable equilibrium point (Methods).
19
10
(strong)
tolerance
Figure 2: Species architectural characteristics and tolerance to change. The figure shows the species’
tolerance (node color) of an increase in mutualism for each of the 80 plants and 97 pollinators belonging
to one network located in the high temperate Andes of central Chile.15 The darker the color, the
higher the change in the strength of mutualistic interaction sustained by each species before becoming
extinct and, in turn, the stronger its tolerance. In this example, the effect of global change is assumed
to increase the strength of mutualistic interaction. The system is simulated with a small mutualistic
trade-off (Methods). We also show two key architectural properties of species: degree (node size) and
contribution to nestedness (node position). Each symbol corresponds to one species or node and links
correspond to the mutualistic interaction between plants (top) and pollinators (bottom). Interestingly,
this figure reveals that some specialist species can be more tolerant than generalist species (far right).
20
degree
contribution to nestedness
1.0
none
0.5
0.0
nestedness
e
d = 1.18
small
0.0
−0.5
−1.0
−1.0
−0.5
0.0
0.5
1.0
d = 1.16
c
f
d = 0.33
−0.5
−1.0
−1.0
−0.5
0.0
large
0.5
1.0
d = 0.89
Mutualistic trade−off
b
●
0.5
1.0
d = 1.56
−0.5
0.0
−0.5
degree
0.5
1.0
−1.0
●
0.0
Correlation with species tolerance to change
d
Decreasing mutualism
−1.0
1.0
d = 0.8
0.5
a
Increasing mutualism
#1−59
#1−59
#1−59
#1−59
Network
Figure 3: Species tolerance to change in large ecological networks. For each of the 59 networks (bars), the
figure shows the Spearman rank correlation of animals’ tolerance to change with degree (orange/left bars)
and contribution to nestedness (blue/right bars). A-C correspond to animals’ tolerance to moving from
a weak to a strong mutualism, while D-F correspond to their tolerance to moving from a strong to a weak
mutualism (Methods). Solid bars correspond to correlations that are significantly (p < 0.05) different
from zero. Correlations are calculated for different gradients of mutualistic trade-offs: A,D, B,E, and
C,F represent none, small, and large mutualistic trade-off, respectively (Methods). The figure also shows
the ratio d of the correlation norms between contribution to nestedness and degree (Methods). Plants’
correlations are significantly similar to animals’. The figure reveals that in order to estimate species’
tolerance, first, one needs to identify the correct direction of change and the mutualistic trade-offs in the
system.
21